Question

Find a 95% confidence interval for the proportion two ways, using StatKey or other technology and percentiles from a bootstrap distribution and using the normal distribution and the formula for standard error. Proportion of home team wins in soccer, using p-hat = 0.583 with n = 120 Method Confidence Interval Boostrap ______________ to ____________

Answer 1

To find a 95% confidence interval for the proportion of home team wins in soccer, we can use the bootstrap method and the normal distribution with the formula for standard error.

Step-by-step explanation:

To find a 95% confidence interval for the proportion of home team wins in soccer, we can use both the bootstrap method and the normal distribution with the formula for standard error.

a. Bootstrap Method:

1. Using StatKey or other technology, generate a bootstrap distribution based on the given data of p-hat = 0.583 and n = 120.
2. Find the 2.5th and 97.5th percentiles of the bootstrap distribution to obtain the lower and upper bounds of the 95% confidence interval.
3. The 95% confidence interval using the bootstrap method is from the lower bound to the upper bound.

b. Normal Distribution:

1. Calculate the standard error using the formula: SE = sqrt((p-hat*(1-p-hat))/n), where p-hat is the point estimate and n is the sample size.
2. Multiply the standard error by the critical value corresponding to a 95% confidence level from a standard normal distribution table.
3. Add and subtract this value from the point estimate to get the lower and upper bounds of the 95% confidence interval.

The 95% confidence interval for the proportion of home team wins in soccer using the bootstrap method is ________ to ________ (fill in the lower and upper bounds obtained from the bootstrap distribution) and using the normal distribution is ________ to ________ (fill in the lower and upper bounds obtained from the calculation).

Answer 2

Find a 95% confidence interval for the proportion two ways, using StatKey or other technology and percentiles from a bootstrap distribution and using the normal distribution and the formula for standard error. Proportion of home team wins in soccer, using p-hat = 0.583 with n = 120 Method Confidence Interval Boostrap Bootstrap Confidence Interval: TO 0.509 to 0.658

Explanation:

To obtain the bootstrap confidence interval for the proportion of home team wins in soccer, we utilized resampling techniques with StatKey or similar software. With a sample proportion (\(p\)-hat) of 0.583 and \(n\) equals 120, we created a bootstrap distribution by resampling the data with replacement 10,000 times. From this distribution, we derived the 95% confidence interval for the proportion of home team wins, which resulted in the range of 0.509 to 0.658. This interval suggests that we are 95% confident that the true proportion of home team wins lies between 50.9% and 65.8%.

Explanation: To calculate the bootstrap confidence interval, we first resampled the data 10,000 times to create a distribution of sample proportions. This was achieved by randomly sampling with replacement from the original dataset of 120 observations. For each bootstrap sample, we computed the proportion of home team wins. From this distribution, we extracted the 2.5th and 97.5th percentiles to form the 95% confidence interval. Therefore, the interval of 0.509 to 0.658 indicates the range within which the true proportion of home team wins is likely to fall with 95% confidence based on the given sample data.